The hierarchy of Davydov’s Ansätze: From guesswork to numerically “exact” many-body wave functions

Abstract

This perspective presents an overview of the development of the hierarchy of Davydov's Ans\"{a}tze and a few of its applications in many-body problems in computational chemical physics.Davydov's solitons originated in the investigations of vibrational energy transport in proteins in the 1970s.Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in the Holstein systems.The multiple Davydov Ans\"{a}tze have been proposed, with the increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian,the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic propertiesusing Davydov's Ans\"{a}tze as its trial states.A quantity proven to disappear for large multiplicities,the Ansatz relative deviation is introduced to quantify how closely the Schr\"{o}dinger equation is obeyed.Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology,applications of Davydov's Ans\"{a}tze are made to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, yielding accurate dynamic and spectroscopic properties of the many-body systems.

Thermo-Field Dynamics Approach to Photo-induced Electronic Transitions Driven by Incoherent Thermal Radiation

“…Several techniques have been developed which can, at least in principle, overcome what has been termed the curse of dimensionality. 67,68 Several methodologies can nowadays be used to accurately describe the quantum dynamics of a system with a large number of DoFs, among which Multi-Configuration Time-Dependent Hartree (MCTDH), 68 Multiple-Davydov's Ansaẗze (MDA), 69 hierarchical equations of motion (HEOM), 47,54,70 Quasi-adiabatic path integrals (QUAPI), 71 and the wide family of Tensor-Network representations 72,73 are probably the most representative. Here, we employ the so-called Tensor Train (TT) format (or Matrix Product States (MPS) in the physics literature) to efficiently represent the vibronic wave function.…”

“…Since the introduction of the tilde space doubles the number of nuclear degrees of freedom (DoF), and since a thermal environment can be realistically mimicked only using hundreds of DoFs, especially if the temperature is very high, as in the case of a thermal radiation field, it is essential to use a methodology suitable to treat a large number of dynamical variables. Several techniques have been developed which can, at least in principle, overcome what has been termed the curse of dimensionality . , Several methodologies can nowadays be used to accurately describe the quantum dynamics of a system with a large number of DoFs, among which Multi-Configuration Time-Dependent Hartree (MCTDH), Multiple-Davydov’s Ansätze (MDA), hierarchical equations of motion (HEOM), ,, Quasi-adiabatic path integrals (QUAPI), and the wide family of Tensor-Network representations , are probably the most representative. Here, we employ the so-called Tensor Train (TT) format (or Matrix Product States (MPS) in the physics literature) to efficiently represent the vibronic wave function.…”

Dynamics of dissipative Landau-Zener transitions in an anisotropic three-level system

“…Inspired by numerically "exact" solutions to a two-site problem with short-range coupling to Einstein phonons [56], Shore and Sander pioneered the multiple Davydov ansätze when they experimented with a trial wave function with two Gaussians as its phonon component, an early forerunner of the multiple Davydov ansätze. In a further development of this method, the multiple Davydov ansätze have been applied extensively to a variety of many-body problems in physics and chemistry by Zhao and co-workers [57][58][59][60][61][62][63][64]. The multiple Davydov ansätze also have two variants, multi-D 1 Ansatz and multi-D 2 Ansatz.…”

Photon‐Assisted Landau Zener Transitions in a Tunable Driven Rabi Dimer Coupled to a Micromechanical Resonator

“…In contrast, the Davydov Ansätze combined with the timedependent variational principle treat all the DOFs on an equal footing and can capture the wave function propagation for all the DOFs. [50,51] As linear superpositions of the single Davydov trial states, [52,53] the multiple Davydov Ansätze have been proposed by Zhao and co-workers, [54,55] and in principle can provide a numerically exact solution to the Schrödinger equation with a sufficiently large multiplicity. [54,55] Thanks to their outstanding numerical accuracy and efficiency, the multiple Davydov Ansätze have been extensively applied in describing states of various complex many-body systems.…”